Well-Posedness and Smoothing Effect for Nonlinear Dispersive Equations-猫眼课题宝

DOI：

发表时间：

2018

期刊：

影响因子：

通讯作者：

中科院分区：

其他

文献类型：

作者：

Y. Tsutsumi

其中α是一个真实的常数，2α/3 ε ∈ Z，T > 0。在（1）中，除了α之外，所有参数都被归一化。方程（1）作为非线性脉冲传播现象的数学模型出现在物理学的各个领域，特别是在非线性光学中（见[54]，[27]和[1]）。到目前为止，方程（1）没有三阶导数，即三次NLS方程吸引了许多数学和物理的兴趣。近年来，随着超短脉冲在光子晶体光纤中的重要性，非线性光学的理论和实验物理学家越来越关注方程（1）中三阶色散的作用。从偏微分方程理论的观点来看，非线性发展方程（1）-（2）的柯西问题的适定性问题是最基本的问题之一。柯西问题被称为局部（分别）。全局）适定，如果以下三个性质成立：（i）局部（分别为整体）解的存在性，（ii）解的唯一性，（iii）解对初始数据的连续依赖性。我们将局部适定性和全局适定性分别称为（LWP）和（GWP）。Caucy问题如果不是适定的，就称为不适定。许多数学家一直在研究在什么样的最大空间中，非线性发展方程的柯西问题是适定的。特别是近二十年来，非线性色散方程的研究取得了很大的进展。本文以方程（1）为例，解释作者与Nobu Kishimoto（RIMS，京都大学）、Tomoyuki Miyaji（Meiji Inst.Adv.Stud.Math.Sci.，明治大学），Tadahiro Oh（爱丁堡大学）和Nikolay Tzvetkov（切尔吉-庞特大学）。首先，在第2节中，我们考虑在H中求解柯西问题（1）-（2），其中s < 0，其元素可能不是函数而是分布。当我们处理由分布组成的空间时，我们立即遇到的问题是非线性项如何有意义，因为分布的乘积不一定是定义良好的。1993年，Bourgain [4]提出了具有Fourier限制范数的所谓的Fourier限制方法。傅立叶限制范数及其变体成功地捕获了非线性振荡的特定特征，并且已被应用于描述非线性波动现象的许多非线性演化方程（参见，例如，[4]，[13]，[18]，[22]-[26]，[32]，[34]-[36]，[40]，[42]，[44]-[49]，[57]-[60]）。傅里叶限制法也为非线性相互作用的研究提供了新的数学视角

where α is a real constant with 2α/3 ̸∈ Z and T > 0. In (1), all the parameters are normalized except for α. Equation (1) appears as a mathematical model for nonlinear pulse propagation phenomena in various fields of physics, especially in nonlinear optics (see [54], [27] and [1]). So far, equation (1) without the third order derivative, that is, the cubic NLS equation has attracted much mathematical and physical interest. Recently, as the ultra-short pulse has become important in the photonic crystal fiber, an increasing attention among theoretical and experimental physicists in nonlinear optics has been paid to the role of the third order dispersion in equation (1). From a viewpoint of the PDE theory, the well-posedness issue of the Cauchy problem for nonlinear evolution equations such as (1)-(2) is one of the most fundamental problems. The Cauchy problem is said to be locally (resp. globally) well-posed if the following three properties hold: (i) local (resp. global) existence of solutions, (ii) uniqueness of solutions, (iii) continuous dependence of solutions on initial data. We refer to the local and the global well-posedness as (LWP) and (GWP), respectively. The Caucy problem is said to be ill-posed if it is not well-posed. Many mathematicians have been studying what is the largest space where the Cauchy problem of a nonlinear evolution equation at hand is well-posed. Especially, there has been a great progress in nonlinear dispersive equations for the last two decades. In this note, we take equation (1) as an example to explain recent results obtained by the author in collaboration with Nobu Kishimoto (RIMS, Kyoto University), Tomoyuki Miyaji (Meiji Inst. Adv. Stud. Math. Sci., Meiji University), Tadahiro Oh (The University of Edinburgh) and Nikolay Tzvetkov (University of Cergy-Pontoise). First, in Section 2, we consider solving the Cauchy problem (1)-(2) in H for s < 0, elements of which may not be functions but distributions. When we work with the space consisting of distributions, the problem we immediately meet with is how the nonlinear term can make sense, because the product of distributions is not necessarily well-defined. In 1993, Bourgain [4] presented the so-called Forier restriction method with the Fourier restriction norm. The Fourier restriction norm and its variants succeeded in capturing specific features of nonlinear oscillations and have been applied to many nonlinear evolution equations describing nonlinear wave phenomena (see, e.g., [4], [13], [18], [22]-[26], [32], [34]-[36], [40], [42], [44]-[49], [57]-[60]). The Fourier restriction method also led a new mathematical insight into the nonlinear interaction

Well-Posedness and Smoothing Effect for Nonlinear Dispersive Equations

文献摘要

相似文献